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Genericity: a Measurehtheoretic Analysis Genericity: A Measure–Theoretic Analysis Massimo Marinacci Draft of December 1994y 1 Introduction In many results of Mathematical Economics the notion of smallness plays a crucial role. Indeed, it often turns out that desirable properties do not hold in all the environment of interest, but only in some of its parts. For example, this is usually the case in the di¤erentiable approach to General Equilibrium (see e.g. [4]). The purpose of the analysis then becomes to prove that the desired property fails only in a negligible part of the environment. This is the so called generic point of view (see e.g. [14, pp. 316- 319]). In a similar approach it is crucial to make precise what is meant by terms like negligible and generic. Two notions of smallness are available. On one hand, nowhere dense subsets and meager (often called …rst category) subsets have been regarded as small in a topological sense. To help intuition, observe that a nowhere dense subset of the real line is a subset full of holes. On the other hand, from a measure-theoretic viewpoint the notion of small is represented by nullsets. Consequently, quoting [15, p.4], ”it is natural to ask whether these notions of smallness are related”. This is the purpose of the study of residual measures, i.e. Borel measures that give measure zero to all meager sets. With respect to a residual measure, meager subsets are small not only in a topological sense, but also in measure–theoretic one. In this paper we will investigate under which conditions these measures exist. In particular, it will turn Department of Economics, Northwestern University. I wish to thank Keith Burns and Itzhak Gilboa for helpful comments. Financial support from Ente Einaudi is gratefully acknowledged. yFirst draft: February 1993. 1 out that in many interesting spaces these measures do not exist. This means that in such spaces for any given …nite Borel measure there always exists a meager subset A such that (A) > 0; i.e. a topologically negligible set which is not negligible from a measure-theoretic standpoint. A similar situation already occurs in Euclidean spaces. In fact, in these spaces genericity analysis integrates topological and measure-theoretic considerations. For example, by considering as negligible the sets whose closure has Lebesgue measure zero. However, in Euclidean spaces the topological and measure-theoretic notions employed have usually a strong intuitive basis, and there is a natural way to integrate them. This is not the case in abstract spaces, where the analysis is usually based on the topological notions of smallness. Our results show that in many spaces of interest it is impossible to have a measure-theoretic support for the topological approach. In other words, there is no Borel measure for which all the topologically negligible sets get measure zero. Stronger results have been obtained for nonatomic measures by Marczewski and Sikorski [12] and [13]. In particular, they have proved that given any nonatomic …nite Borel measure de…ned on a metric spaces with a basis whose cardinal is nonmeasurable (see below; separable metric spaces are an example) it is always possible to decompose the space in two disjoint subsets, one meager, the other one of measure zero. The results presented here are weaker, but hold for any …nite Borel measure, not necessarily nonatomic. As an aside, it is worth noting that the results of Marczewski and Sikorski are im- portant for the analysis of Nash re…nements in normal form games with in…nite strategy spaces. Here the problem is to …nd the appropriate in…nite counterpart of completely mixed strategies, on which re…nements for …nite games are based. A natural candidate are the Borel measures with full support, i.e. measures assigning strictly positive mass to every nonempty open subset. In Simon and Stinchcombe [19] is developed an ap- proach to in…nite games re…nements based on full support Borel measures. However, the results of Marczewski and Sikorski show that for every full support nonatomic mea- sure on a metric space with a basis whose cardinal is nonmeasurable, there exists a partition A; Ac such that A is meager and (Ac) = 0. In other words, even though f g puts strictly positive measure to all open sets, there exists a topologically large set like Ac which gets zero measure under . As nonatomic measures are an important class of mixed strategies, this shows that full support measures are a much weaker notion 2 than completely mixed strategies in …nite spaces. Besides genericity analysis, residual measures are interesting in the theory of Boolean algebras. Indeed, let A be a Boolean -algebra and P a probability measure (not nec- essarily strictly positive) de…ned on it. By the Loomis-Sikorski representation theo- rem, to the pair (A; P ) can be associated a measure space characterized by a residual measure. Using techniques of this type it is possible to derive some results about prob- ability measures de…ned on Boolean algebras from propositions established for residual measures. The paper is organized as follows. Section 2 contains some preliminary notions. Section 3 gives existence results concerning residual measures. In section 4 the results of section 3 are used in order to obtain some non-existence results for probability measures de…ned on Boolean algebras. Finally, in section 5 the constructions of the previous sections are used to get a result about the ranges of probability measures on Boolean algebras. We follow the notation of [10]. This implies, for instance, that lower and upper cases will usually denote, respectively, sets and Boolean algebras. 2 Preliminary notions 2.1 Measures Let X be a topological space and let cl(a) and int(a) denote, respectively, the closure and the interior of a subset a X. A subset a is nowhere dense if int (cl(a)) = ?, i.e. if the closure of a has empty interior. Nowhere dense subsets have no interior points. A subset a is meager (or of the …rst category) if it can be represented as a countable union of nowhere dense subsets. They are the subsets of X which can be approximated by nowhere dense subsets. We now de…ne some classes of Borel probability measures. De…nition 1 A Borel probability measure P de…ned on a topological space X is regular if for all Borel subsets a we have P (a) = sup P (c): c aand cclosed = inf P (v): a vand vopen : f g f g 3 De…nition 2 A Borel probability measure P de…ned on a topological space X is - additive if whenever g is a net of open subsets such that g g for , then f g P g = sup P (g ): ! [ Let M denote the -ideal of all meager Borel subsets. Next we de…ne residual measures. De…nition 3 A Borel probability measure P de…ned on a topological space X is resid- ual if P (a) = 0 for all a M. 2 Given a probability measure P on a …eld A, a subset a A is called a P -atom if: 2 (i) P (a) > 0; (ii) if b a and b A, then either P (b) = 0 or P (a b) = 0: 2 n De…nition 4 A Borel probability measure P de…ned on a topological space X is nonatomic if there are no P -atoms. In the sequel we make use also of the notion of nonmeasurable cardinal. Indeed, suppose that on the power set of a space X there exists a di¤use probability measure , i.e. a measure giving measure zero to all singletons. In such a case, the cardinal number of X is called measurable. Instead, a cardinal is called nonmeasurable if the @ previous property does not hold for all spaces of cardinality . @ Now we turn to Boolean algebras. For basic notions and results we refer to [10]. We begin with a de…nition. A bit of notation: 1 is the unit element of A, while n1=1 an is the supremum of an 1 in A. f gn=1 P De…nition 5 A probability measure P on a Boolean algebra A is a real-valued function such that: (i) P (a) 0 for all a A; 2 (ii) P ( 1 an) = 1 P (an) whenever an 1 A is a set of pairwise disjoint n=1 n=1 f gn=1 elements for which 1 a exists; P P n=1 n P 4 (iii) P (1) = 1: It is worth noting that P has not been de…ned as a strictly positive measure (i.e. it is not required P (a) = 0 i¤ a is the zero element). It is important to distinguish the previous de…nition from the usual de…nition of probability measures on …elds of subsets. Let A be a …eld of subsets. A real–valued set function P on A is countably additive if: (iii’) P ( 1 an) = 1 P (an) whenever an 1 A is a set of pairwise disjoint n=1 n=1 f gn=1 elements such that 1 an A. S P n=1 2 S Condition (iii’)is quite di¤erent from condition (iii) in de…nition 5, as it was stressed in [8]. In fact, 1 an, the supremum of an 1 in A, can exists even when 1 an is n=1 f gn=1 n=1 not in A. Instead, if 1 an A, then 1 an = 1 an. Therefore, if a set function P n=1 2 n=1 n=1 S satis…es (iii’), then it satis…es (iii), while a set function which satis…es (iii’)can fail to S S P satisfy (iii).
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